# ◈ SYMMETRY ◈

Background

All our lives, we have been exposed to symmetry. In general, we use the term “symmetrical” to describe an object that is the same on one side as the other. A snowflake is symmetrical by this definition; it can be cut in half with both sides being the same. However, this is only the tip of the iceberg when it comes to symmetry in a chemical context; symmetry encompasses a lot more than just two sides matching up. Rotational axes and mirror planes of all kinds are combined to determine a molecule’s“point group” (we know, lots of chemistry jargon right there, we will explain, we promise). These designations help scientists to understand the shape of different molecules, and can give insight as to how those chemicals work. So, let’s look at what a chemist would say are the symmetrical properties of a snowflake:

In order to get a better idea about how symmetry works in chemistry, we have to start with the basics. Think of a triangle. Better yet, go get a piece of paper and cut one out (make sure all the sides are the same length; 60° angles now!). When you look at it, all the points are the same. Think about ways you can turn it and flip it so that if you were to show it to a friend, manipulate it, and show it to a friend again, they wouldn’t know what happened. In order to more easily follow all the operations, put a toothpick through the center of your triangle. A planar (flat) triangle has D3h symmetry, which is comprised of the following parts: Figure 2. Triangle rotating on a C3 axis.

Every ⅓ rotation along the toothpick yields the triangle’s original conformation. This is called a C3 axis (3 for the number of rotations you can do to yield the same, indiscernible shape). For the purposes of this demonstration, we included colored dots to make the rotations more easily discernible–they otherwise ruin the perfect symmetry! Figure 3. Triangle rotating on a C2 axis.

It also has three unique C2 axes extending from each point to the opposite base. You can think about this by placing the toothpick on top of the triangle so it divides it in half. If you flip it over, the triangle is the same! Each C2 axis runs perpendicular to the main C3 axis, classifying this shape as having D symmetry.

In addition to a C2 axis, this toothpick represents a σv, or vertical plane of symmetry (think mirror). You can also represent this by folding your triangle in half. There are three of these planes.

In addition, there is one final plane of symmetry. Can you think of it? There is a plane of symmetry that divides your sheet of paper in half. Yup. Even though the paper is very very very thin, there is still one plane of symmetry that runs flat through the sheet of paper! This is called a σh, or horizontal plane of symmetry. This gives us our “h” in D3h.

In total, there are six different ways you can orient the same triangle to get six distinct, but identical shapes.

So what about Mr. Wilson Bentley’s snowflake? Lets take what we’ve learned and apply it. Figure 4. Snowflake showing all sorts of symmetry operations.

To explore more symmetry, check out the Otterbein Symmetry Gallery.

Also, to try and walk through your own shape, check out this really confusing symmetry chart.

Object D4h: Why it’s cool

So now let’s apply what we’ve learned to our sculpture. Can you see that there is a rotation axis running through from top to bottom? By rotating a ¼ turn (90º) along this axis, we observe that the same conformation is retained. This is a C4 axis because you can turn the shape four times around this axis and end up with the same orientation.  Figure 5. Condensed view of Object D4h with C4 axis down in two perspectives

Now let’s shift our attention to the central region. If we draw an axis through the center of the orb and 3 more at 45° intervals, we find four C2 axis. Turn the shape 180° around these axis, do you see that you get the same orientation? Figure 6. A single C2 axis running through the center of Object D4h.

Finally, imagine if you were to slice the object in half through the center orb, forming a horizontal reflection plane. You get a mirror image on each side! Combining all these symmetry elements: you get a classification of D4h. Sound familiar?